Wide Effectiveness of a Sine Basis for Quantum-Mechanical Problems in 𝑑 Dimensions

Table 3

Approximation of the energy levels of the harmonic oscillator in dimensions 𝑑=3,4,5. 𝑛 represents the energy state, 𝐸 is the exact solution for the energy given by (12), and 𝜀 is the upper bound to 𝐸 obtained using the present variational analysis. The eigenvalues 𝜀_𝑛 of H are minimized over 𝐿.

𝑑	ℓ	𝑛	𝐸	𝜀	𝐿
3		1	3	3.00000000	6.00
	0	5	19	19.00000001	7.00
		10	39	39.00000001	8.75
		1	5	5.00007348	4.50
	1	5	21	21.00167944	6.25
		10	41	41.00907276	7.75
		1	7	7.00000000	6.00
	2	5	23	23.00000001	7.75
		10	43	43.00000001	9.25
		1	9	9.00000001	6.00
	3	5	25	25.00000076	7.25
		10	45	45.00002070	8.50
4	0	1	4	4.00073469	4.25
		5	20	20.00745550	6.00
		10	40	40.02454449	7.50
	1	1	6	6.00000262	5.00
		5	22	22.00011370	6.50
		10	42	42.00094014	8.00
	2	1	8	8.00000002	6.00
		5	24	24.00000248	7.25
		10	44	44.00004592	8.50
	3	1	10	10.00000000	6.00
		5	26	26.00000008	7.50
		10	46	46.00000274	8.75
5	0	1	5	5.00007348	4.50
		5	21	21.00167944	6.25
		10	41	41.00907276	7.75
	1	1	7	7.00000000	6.00
		5	23	23.00000001	7.75
		10	43	43.00000001	9.25
	2	1	9	9.00000001	6.00
		5	25	25.00000076	7.25
		10	45	45.00002070	8.50
	3	1	11	11.00000001	6.00
		5	27	27.00000000	8.00
		10	47	47.00000001	10.00